Sunday, February 20, 2011

Finding the roots of polynomials defined by infinite primes

Infinite primes identifiable as analogs of free single particle states and bound many-particle states of a repeatedly second quantized supersymmetric arithmetic quantum field theory correspond at n:th level of the hierarchy to irreducible polynomials in the variable Xn which corresponds to the product of all primes at the previous level of hierarchy. At the first level of hierarchy the roots of this polynomial are ordinary algebraic numbers but at higher levels they correspond to infinite algebraic numbers which are somewhat weird looking creatures. These numbers however exist p-adically for all primes at the previous levels because one one can develop the roots of the polynomial in question as powers series in Xn-1 and this series converges p-adically. This of course requires that infinite-p p-adicity makes sense. Note that all higher terms in series are p-adically infinitesimal at higher levels of the hierarchy. Roots are also infinitesimal in the scale defined Xn. Power series expansion allows to construct the roots explicitly at given level of the hierarchy as the following induction argument demonstrates.

At the first level of the hierarchy the roots of the polynomial of X1 are ordinary algebraic numbers and irreducible polynomials correspond to infinite primes. Induction hypothesis states that the roots can be solved at n:th level of the hierarchy.

At n+1:th level of the hierarchy infinite primes correspond to irreducible polynomials

Pm(Xn+1)= ∑s=0,...,m ps Xsn+1 .

The roots R are given by the condition

Pm(R)=0 .

The ansatz for a given root R of the polynomial is as a Taylor series in Xn:

R= ∑ rkXnk ,

which indeed converges p-adically for all primes of the previous level. Note that R is infinitesimal at n+1:th level. This gives

Pm(R)=∑s=0,...,m ps (∑ rkXnk)s=0 .

The polynomial contains constant term (zeroth power of Xn+1 given by

Pm(r0)=∑s=0,...,m pr r0s .

The vanishing of this term determines the value of r0. Although r0 is infinite number the condition makes sense by induction hypothesis.

One can indeed interpret the vanishing condition

Pm×m1(r0)=0

as a vanishing of a polynomial at the n:th level of hierarchy having coefficients at n-1:th level. Here m1 is determined by the dependence on infinite primes of lower level expressible in terms of rational functions. One can continue the process down to the lowest level of hierarchy obtaining m×m1...×mk:th order polynomial at k:th step. At the lowest level of the hierarchy one obtains just ordinary polynomial equation having ordinary algebraic numbers as roots.

One can expand the infinite primes as a Taylor expansion in variables Xi and the resulting number differs from an ordinary algebraic number by an infinitesimal in the multi-P infinite-P p-adic topology defined by any choice of n-plet of infinite-P p-adic primes (P1,...,Pn) from subsequent levels of the hierarchy appearing in the expansion. In this sense the resulting number is infinitely near to an ordinary algebraic number and the structure is analogous to a completion of algebraic numbers to reals. Could one regard this structure as a possible alternative view about reals remains an open question. If so, then also reals could be said to have number theoretic anatomy.

If one has found the values of r0 one can solve the coefficients rs, s>0 as linear expressions of the coefficients rt, t<s and thus in terms of r0.

The naive expectation is that the fundamental theorem of algebra generalizes so that that the number of different roots r0 would be equal to m in the irreducible case. This seems to be the case. Suppose that one has constructed a root R of Pm. One can write Pm(Xn+1) in the form

Pm(Xn+1)= (Xn+1-R) × Pm-1(Xn+1) ,

and solve Pm-1 by expanding Pm as Taylor polynomial with respect to Xn+1-R. This is achieved by calculating the derivatives of both sides with respect to Xn+1. The derivatives are completely well-defined since purely algebraic operations are in question. For instance, at the first step one obtains Pm-1(R)=(dPm/dXn+1)(R). The process stops at m:th step so that m roots are obtained. At lower levels similar branching occurs just as it occurs for polynomials of several variables.

What is remarkable that the construction of the roots at the first level of the hierarchy forces the introduction of p-adic number fields and that at higher levels also infinite-p p-adic number fields must be introduced. Therefore infinite primes provide a higher level concept implying real and p-adic number fields. If one allows all levels of the hierarchy, a new number Xn must be introduced at each level of the hierarchy. About this number one knows all of its lower level p-adic norms and infinite real norm but cannot say anything more about them. The conjectured correspondence of real units built as ratios of infinite integers and zero energy states however means that these infinite primes would be represented as building blocks of quantum states and that the points of imbedding space would have infinitely complex number theoretical anatomy able to represent zero energy states and perhaps even the world of classical worlds associated with a given causal diamond.

9 Comments:

What led to the introduction of p-adic numbers was the following observation (around 1995). p-Adic thermodynamics for conformal scaling generator representing mass squared in super-conformal invariant system and taking same role as energy in ordinary thermodynamics allows to understand the basic mass scales of elementary particle physics in terms of CP_2 size scale which is about 10^4 Planck lengths.

It turned out that p-adic thermodynanamics allows to predict particle masses within accuracy of percent if one accepts that p-adic prime is near integer power of two and allows this integer as characterizer of particle. The mass scale of particle is exponentially sensitive to this integer: therefore there is no possibility of fitting and the success is really stunning.

This prime is large: for electron it is Mersenne prime M_127=2^127-1 =about 10^38, which is the largest Mersenne not defining completely super-astrophysical p-adic length scale. This also guarantees that the numerical errors in the calculation taking into account only two lowest term in the expansion in powers of 1/p is about 10^(-76)! This is certainly the perturbative expansion with fastest convergence in the history of theoretical physics!

I have now continued 15 yearst my attempts to communicate this breakthrough but without success. This is one reason for why I talk about Masters of the Universe attitude. Arrogance can make even the brightest scientist a complete idiot.

p-Adic physics is now a key part of quantum TGD and also of TGD inspired theory of consciousness.

p-Adic length scale hierarchy makes sense also in macro scales. The length scale from cell membrane thickness to cell nucleus size is mathematically miraculous since as many as 4 Gaussian Mersennes (complex primes of form (1+i)^k-1 are in this range and correspond to k=151,157,163,167. k=151 corresponds to 10 nm defining cell membrane thickness. Therefore it seems that life is extremely special from p-adic point of view.

.1 seconds which is fundamental time scale in living matter (consider for instance alpha rhythm) corresponds to secondary p-adic time scale associated with Mersenne prime M_127 associated with electron. Quite generally, the secondary p-adic time scales assigned with elementary particles would appear as macroscopic time scales meaning a completely unexpected connection with microscopy and macroscopy.

These are just the two most glaring examples in biology. The hypothesis makes sense also in astrophysics and leads to quite fascinating proposals. Here testing would require more advanced methods than available to me.

Short range chaos combined with long range correlations is the basic dynamical signature. Shnoll's findings about deviations of various rates (such as nuclear decay rates) from expected and varying periodically in astrophysical time scales led for some time ago to a very general deformation of probability distributions by introducing p-adic prime p and quantum phase q= exp(i2pi/m) so that deformation is parameterized by p and m.

For instance, Poisson distribution P(n) becomes many peaked and periods are predicted to occur periodically in n with period which is power of some prime. This is a very general testable prediction applying to all distributions. You can find an article about this in Prespacetime Journal December 2010.

I ask because I'm inspired by your postulation of p-adic physics/numbers and consciousness. I truly wonder to what extent p-adic numbers can be applied...You say "short range chaos with long range correlations." A model to predict human behavior?

In consciousness theory p-adic physics would serve as a correlate for cognition and perhaps also intentionality: thought bubbles;). The transformation of intention to action would mean quantum jump replacing p-adic space-time sheet representing intention to real space-time sheet representing the action. Zero energy ontology allows to formulate this mathematically since net quantum numbers vanish and do not produce trouble. Essential is also the generalization of the notion of number by fusing reals and p-adics along common rationals to a larger structure.

p-Adic topology and real topology differ dramatically. What is infinitesimal p-adically is infinite in real sense. p-Adic smoothness means long range correlations in real sense and vice versa if p-adic and real space-time sheets have large enough number of common points. The success of p-adic mass calculations could be indeed interpreted in terms of effective p-adic topology applying to real space-time sheets (or their discretizations) and would suggest that cognition is present even at elementary particle scales.

p-Adic and real smoothness would pose very strong constraints on physics on UV and IR: the success of p-adic mass calculations would be excellent example of the ensuing predictive power. In M-theory strings give only UV constraints and this leads to the landscape problem.

Short range chaos and long range correlations could indeed serve as a model for human behavior. Our intentional behavior is not predictable in short time scales but is so in long time scales.

The statistical aspects of human behavior might be predictable. There is a thesis work claiming that distributions related to market behavior have similar many-peaked structure as observed by Shnoll in totally different systems.

About Me

I am a Finnish theoretical physicist. For last 37 years Topological Geometrodynamics has been both the passion and mission of my life. TGD is a noble attempt to construct a theory of everything, not forgetting consciousness. I have four children, who have brought a lot of happiness to my life. I live in Hanko, a small seaside town in southern Finland. I love almost all kinds of music but if I had to give just one name I would have difficulties in deciding between Chopin and Beethoven.

The 37 years with TGD have produced an enormous amount of material covering basic
TGD as a mathematical theory, the applications of TGD ranging from Planck length scale
to cosmology, and TGD inspired theory of consciousness and of living matter as a macroscopic
quantum system. I have organized this material at my homepage as online books and articles.